Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

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How to compute $ \int e^{-st} \sin(2t) dt $

Wolfram Alpha shows me the result of $ \int e^{-st} \sin(2t) dt $ . However it doesn't let me see the step to step solution. Then I tried to do this by hand as the solution didn't look "too difficult", however couldn't do it. So can someone show…
Imago
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Is there an alternative way to solve this integral?

I was given the integral $$\int \frac{2}{e^{-x}+1}dx$$ Here is my method to get the (correct) solution: $$\int \frac{2}{e^{-x}+1}dx$$ $$=2\int \frac{1}{e^{-x}+1}dx$$ $$=2\int \frac{e^xe^{-x}}{e^{-x}+e^xe^{-x}}dx$$ $$=2\int\frac{e^x}{1+e^x}dx$$ Let…
HDE 226868
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Limit integration

I have: $$ F(a)=\int_0^a(x^2+1)e^{-x/2} dx $$ I have done the integration: $$ \int(x^2+1)e^{-x/2}=-2e^{-x/2}(x^2+4x+9)+C$$ What is (if possible): $$ lim_{a \to \infty} F(a)$$ I tried: $$ (-2e^{-a/2}(a^2+4a+9))-(-2e^{-0/2}(0^2+4*0+9)=$$ $$…
peppa
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6D integral with gaussian ansatz

Does anyone know how to handle this integral: $$\int d^3 r \int d^3 r' \phi(\mathbf{r})\phi(\mathbf{r'})\frac{Y^{0}_{2}(\mathbf{r} - \mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|^3}$$ with $$\phi(\mathbf{r}) = \exp\left( -\frac{x^2}{\sigma_{1}^{2}}…
WoofDoggy
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Integration without substitution of $\frac{x^2+3}{x^6\left(x^2+1\right)}$

This is a repost of a question i had written incorrectly earlier. How do I integrate this without substitutions ? $$ \frac{x^2+3}{x^6\left(x^2+1\right)} $$ I got: $$ \frac{1}{x^6}+\frac{2}{x^6\left(x^2+1\right)}, $$ but wasn't able to eliminate…
questioner
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Integration without substitution

How to i integrate this with out substitutions or Partial fraction decomposition ? ($3x^2$+$2$)/[$x^6$($x^2$+1)] I've got to : 2/x^6(x^2+1),but after this i haven't been able to eliminate the 2.
questioner
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intregration without substitution of $x^x \ln x$

How do i integrate this without any substitution, purely algebraically : $$x^x \ln ex$$ I've tried a lot but not have been able to: $$x^x \ (ln x + 1) = \ln x^{x^x} + x^x$$ or $e^{x \ln x}\ln (x+1)$, i've tried all these methods How do proceed…
questioner
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Reduction Formula

If $\displaystyle I_{n}\equiv \int^{\pi/4}_{0} \tan^{n}\left(\,\theta\,\right)\,{\rm d}\theta$ prove that $\displaystyle I_{n} = {1 \over n - 1} - I_{n - 2}$ I tried using integration by parts by first writing $tan\theta$ as $tan\theta^{n-1} …
user140161
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Integrate $\ln(2-x)dx$

I want to learn how to integrate this. If you could show me a step-by-step approach that would be awesome. If you could also point me to some good tutorials on integration that would be icing on the cake. What does $dx$ mean? I know its delta $x$…
gedr
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Evaluate a Nested Integral

According to two questions [1] and [2] asked on this site earlier there exists a nice relation: $$\frac1{n!} \left(\int_{0}^t\mathrm dt \; f(t)\right)^n = \int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\;…
Yrogirg
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Integrate $\int\frac{\cos^2x}{1+\tan x}dx$

Integrate $$I=\int\frac{\cos^2x}{1+\tan x}dx$$ $$I=\int\frac{\cos^3xdx}{\cos x+\sin x}=\int\frac{\cos^3x(\cos x-\sin x)dx}{\cos^2x-\sin^2x}=\int\frac{\cos^4xdx}{1-2\sin^2x}-\int\frac{\cos^3x\sin xdx}{2\cos^2x-1}$$ Let $t=\sin x,u=\cos x,dt=\cos…
RE60K
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Very few elementary functions with elementary antiderivatives

I have heard that most elementary functions don't have elementary antiderivatives. Is there a precise meaning to the previous sentence, and if so, may I see a paper where the precise version of that statement is proven?
user107952
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Computing the integral $\int \frac{u}{b - au - u^2}\mathrm{d}u$

After working on an ODE I find I am needing to solve the integral $$\int \frac{u}{b - au - u^2}\mathrm{d}u$$ Trig subs, banging heads against walls, and sobbing have not yielded a solution. Yet. Could use a hand, thanks.
Andres Salas
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$\int \sqrt{1+\sin ^2 x} dx$ an elliptic integral?

It seems to be an elliptic integral of the second kind, but when $k=i$? This is going by the definition that $E(\theta,k)=\int_{0}^{\theta} \sqrt{1-k^2 \sin^2x}dx$. That seems a bit off. Or is this not one at all due to the indefinite nature of the…
Trogdor
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How to evaluate the following two integral combined with anti-trigonometric function and trigonometric function？

\begin{align*} &\int_0^{\frac{\pi }{3}} {\arccos \frac{{1 - \cos x}}{{2\cos x}}dx} \\ &\int_0^{\frac{\pi }{2}} {\arccos \sqrt {\frac{{\cos x}}{{1 + 2\cos x}}} dx}. \end{align*} A few days ago,my e-friend asked me two integral,but he also doesn't…
Eufisky
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